Problem: Simplify the following expression and state the condition under which the simplification is valid. $q = \dfrac{k^2 - 49}{k + 7}$
Answer: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = k$ $ b = \sqrt{49} = 7$ So we can rewrite the expression as: $q = \dfrac{({k} + {7})({k} {-7})} {k + 7} $ We can divide the numerator and denominator by $(k + 7)$ on condition that $k \neq -7$ Therefore $q = k - 7; k \neq -7$